Monday, May 1, 2017

“Number is the tune to which all things move, and as it were make music; it is in the pulses of the blood no less than in the starred curtain of the sky. It is a necessary concomitant alike of the sharp bargain, the chemical experiment, and the fine frenzy of the poet. Music is number made audible; architecture is number made visible; nature geometrizes not alone in her crystals, but in her most intricate arabesques.”(2)

An early 8th century, repeating tile tessellation from the Umayyad Palace of Khirbat al-Mafjar near Jericho, Palestine. (AramcoWorld, Volume 66, Number 6, November/December 2015, Special Insert)

There is a similarity of patterned design across many Islamic arts and crafts and throughout all Islamic historic periods and across geographic regions. These patterns are found in ceramic tile and pottery designs,

"Star Ushak" Carpet, late 15th century. Woven in the Ushak region of western Turkey, "Star Ushak" carpets were made for regional consumption and for export throughout Europe. (Metropolitan Museum of Art, New York, New York, Gift of Joseph V. McMullan, 1958, Accession number 58.63; http://www.metmuseum.org/toah/works-of-art/58.63/)

“Islamic art comprises geometric prints and tessellations which connect to form infinitely repetitive designs. They are beautiful in their simplicity, creating shapes and patterns that simultaneously satisfy and intrigue the eye of the observer. They invite the observer to contemplate the nature of the relationship between beauty and geometry[... .]”(3)

Detail of a tile panel with a repeating vegetal pattern in the Rüstem Paşa Camii (Rustem Pasha Mosque) in Istanbul built in 1560 by the architect Mimar Sinan. (Photo credit: Michael Padwee, 2011)

Muslim intellectuals believe that "this relationship between beauty and geometry is an expression of the fundamental truths of Tawhid and Mizan, without committing Shirk (idolatry – the unforgivable sin). Tawhid, the ultimate unity of all things, emanating from the Oneness of God, is depicted in the patterns which tessellate into infinity. The fact that these tessellating designs are found to be so pleasing to the eye is evocative of the order and balance of the universe, expressed through the laws of geometry. For the Islamic artists, beauty and truth come together in geometric designs to express the perfection of God.”(4) Thus, the preference for geometry is directly related to the religious essence of Islamic culture. Such forms convey an aura of spirituality, but with no symbolic significance that attacks the beliefs of Islam. They also believe that symmetry (5) evokes a transcendent beauty by being able to stimulate and free the intellect.(6)“There is a certain disregard for scale in Islamic art that derives from this perception. Similar kinds of patterning, for instance, might be found on a huge tile panel or on a bijou ornament. This is because decorative effects, in an Islamic context, are never mere embellishments, but always refer to other, idealised states of being. In this view, scale is almost irrelevant. For similar reasons Islamic ornemanistes (7) usually opted for a-centric arrangements in patterning, avoiding obvious focal points – a preference that resonates with the Islamic perception of the Absolute as an influence that is not ‘centred’ in a divine manifestation (as in Christianity), but whose presence is an even and pervasive force throughout the Creation.”(8)

There are many types of ornamentation that we generally recognize as “Islamic” patterns, yet “the whole range of Islamic patterns represents an amalgam of many different styles, some simply adapted and absorbed from classical sources and from various cultures with which Islam came into contact during its early expansion.”(9) In his article, “Islamic Star Patterns”, A. J. Lee states that “in their simplest form all Islamic geometrical patterns are examples of periodic tilings (or tessellations) of the two dimensional plane, consisting of polygonal areas (10) or cells of various shapes abutting on neighboring cells at lines termed the edges of the tiling, and with three or more cells meeting at points termed the verticies, or nodes, of the tilings.”(11)In 1905 Islamic scholar E. H. Hankin wrote of his research into the creation of Islamic tile patterns, and of the serendipitous discovery of a four-century-old working drawing of an arabesque pattern in a palace of the Mogul Emperor Akbar that led him to his conclusions. Hankin describes four classes of elementary patterns used in Islamic tilings: a) the space to be decorated is divided into squares where the dividing lines are at right angles, such as a chess board, and parts of these squares help form the pattern;

In all Islamic tile patterns, “[the] artist simply uses the full but confined surface of the material in the chosen technique in a particular mathematical structure, thus creating a balanced and often rhythmic image consisting of human, floral or abstract figures in a seamless, pattern-like design. The repetition of this pattern offers a sense of tranquillity, movement or even infinity... .”(13)A.J. Lee uses star patterns to illustrate geometric patterns in the formation of Islamic tessellations.

Drawings of star patterns (clockwise from UL): two early rectilinear star patterns; a colinear link between two 9-pointed stars; a parallel link between two 10-pointed stars; an eighth century pattern with curvilinear 8-pointed stars; 12-pointed stars on a grid of squares; and 12-pointed stars on a triangular grid. (A.J. Lee, “Islamic Star Patterns”, Muqarnas, Vol. 4, 1987, pp. 182-197; http://www.jstor.org/stable/1523103)

“[...The] most typically ‘Islamic’ of all star motifs...is the geometrical rosette... .

12- and 6-rayed geometrical rosettes and a 6-rayed rosette seen as a discrete motif. “The prototype for the general n-rayed rosette almost certainly consisted of a 6-pointed star surrounded by six regular hexagons... . [...The] prototypical 6-rayed rosette seems to have been used for the first time as a distinct motif on the Arab-Ata mausoleum (978) at Tim, in Uzbekistan... .” (A.J. Lee, “Islamic Star Patterns”, Muqarnas, Vol. 4, 1987, pp. 188, 183; http://www.jstor.org/stable/1523103)

Dutch artist Maurits Cornelis Escher (1898-1972) and his wife, Jetta, visited the Alhambra in Granada, Spain in 1922, and again in 1936. On these trips Escher saw Islamic-patterned tile installations first hand, and he was inspired by the patterns he saw. “[Escher] was fascinated by the effect of colour on the visual perspective, causing some motifs to seem infinite–an effect partly caused by symmetry [see endnote 5]. Despite his efforts in the...years [following 1922], Escher failed to understand the principles behind tessellation. Only between 1937 and 1942 [did] he succeed..in doing so, after he had visited the Alhambra for [the] second time... .”(14)

Tessellations, arabesques and calligraphy on a wall of the Myrtle Court in the Alhambra. (Photo credit: By Jebulon - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11234546) “In the Alhambra (14th C), Spain..., geometric pattern is perfectly integrated with biomorphic design (arabesque) and calligraphy. These are the three distinct, but complementary, disciplines that comprise Islamic art. They form a three-fold hierarchy in which geometry is seen as foundational. This is often signified by its use on the floors or lower parts of walls, as shown in the image above.” (Richard Henry, “Geometry – The Language of Symmetry in Islamic Art”, Art of Islamic Pattern, p. 3; http://artofislamicpattern.com/resources/educational-posters/)

“Escher...was fascinated by every kind of tessellation—regular and irregular—and took special delight in what he called ‘metamorphoses,’ in which the shapes changed and interacted with each other, and sometimes even broke free of the plane itself."(15) Escher was very successful at depicting the real world in a 2-dimensional plane as well as at translating the principles of regular division onto a number of 3-dimensional objects such as spheres, columns, and cubes. Also, some of his prints combine both 2 and 3-dimensional images.(16)

Escher experimented with simple repetition of a plane and mirror images to create his own tessellations. "In many of Escher’s tessellations, there is a clear reference to the geometric shapes and principles used in the Islamic art that he saw during his visit [to the Alhambra]. In particular, Escher’s circular tessellations reflect the Islamic principle of eternity, as..., in theory the pattern can [be] repeated infinitely as the circle expands.”(17)

For Escher, “the infinite regular repetition of the tiles in the hyperbolic plane, growing rapidly smaller towards the edge of the circle, [...] allow[ed] him to represent infinity on a two-dimensional plane.” (Hyperbolic tessellation (18): Circle Limit III, 1959; Photo Credit: By Fair Use, https://en.wikipedia.org/w/index.php?curid=7516812) The process of creating hyperbolic drawings by hand was extremely difficult and tiresome for Escher (19) much more so than creating Euclidian tilings. Escher did not have the help of computers, which would have greatly aided him at the time.(20)

Islamic tile patterns were created using geometric grids, and "Escher mastered his skills on geometric grids and used them as the basis for his sketches, later improving them with additional designs, mainly animals such as birds, lions, and reptiles."(21) The following set of drawings illustrates how Escher could modify the geometric grids found in Islamic art to form one type of tessellation:

The tessellation pattern is created by cutting portions of the pattern as in (b) and (d), and mounting them to the correct locations of the pattern considering the rotations and reflections as in (c) and (e). Finally, the pattern is rendered in tile as illustrated in (f). (21)

Another of Escher's tessellations, Metamorphosis II, illustrates the unity of living things, something that informed many of Escher's images. "In many of Escher’s tessellations, not only the patterns, but also the entire scene is inspired by the circle and eternity as in Islamic art. ...In his 'Metamorphosis II,' it is also interesting to see that a closed cycle is formed when the two vertical ends of the picture are joined together."(22)

"Metamorphosis II", Woodcut, 1940, 7.6 in × 153.3 in. (By Official M.C. Escher website., Fair use; http://www.mcescher.com/gallery/switzerland-belgium/metamorphosis-ii/) "[...The] concept of this piece is to morph one image into a tessellated pattern and then slowly alter that pattern eventually to become a new image. The process begins left to right with the word metamorphose (the Dutch form of the word metamorphosis) in a black rectangle, followed by several smaller metamorphose rectangles forming a grid pattern. This grid then becomes a black and white checkered pattern, which then becomes tessellations of reptiles, a honeycomb, insects, fish, birds and a pattern of three-dimensional blocks with red tops. These blocks then become the architecture of the Italian coastal town of Atrani (see Atrani, Coast of Amalfi). In this image Atrani is linked by a bridge to a tower in the water, which is actually a rook piece from a chess set. There are other chess pieces in the water and the water becomes a chess board. The chess board leads to a checkered wall, which then returns to the word metamorphose."(https://en.wikipedia.org/wiki/Metamorphosis_II)

With the help of his study of Islamic art and tessellation patterns, Escher created "exquisite and mind boggling pictures...drawn from the mathematical world of symmetry, topology, transformational geometry, and regular divisions of the plane. [...They] exhibit a rich and artistic talent unrivaled by most. Furthermore, respected scientists have realized that his works are simple illustrations of sophisticated theories[,...some in the field of algebra and higher mathematics, and] Escher has also inspired scientists in their academic studies."(23)

"There is a beauty in discovery. There is mathematics in music, a kinship of science and poetry in the descrip- tion of nature, and exquisite form in a molecule. Attempts to place different disciplines in different camps are revealed as artificial in the face of the unity of knowledge. All literate men are sustained by the philosopher, the historian, the political analyst, the economist, the scientist, the poet, the artisan and the musician." (Glenn T. Seaborg, scientist, Nobel Laureate)

NOTES:

1. I am not a mathematician, but my discussion today necessarily deals with mathematical concepts. I apologize in advance for any lack of explanatory ability on my part in the use of those concepts. For a database of Islamic tile patterns along with explanations, please see http://patterninislamicart.com/about. Also, for a brief explanations of the different types of tessellations, see "Tessellation: The Geometry of Tiles, Honeycombs and M.C. Escher" by Robert Coolman, http://www.livescience.com/50027-tessellation-tiling.html.

￼The left figure is not closed, and the figures in the middle are not made of line segments. The figure on the right is not a polygon, since its sides intersect each other.VertexA vertex of a polygon is a point where two sides come together.

A fundamental characteristic of any polygon is the number of sides it possesses, which is the same as its number of angles."

11. A. J. Lee, p. 183. 12. E. H. Hankin, M.A., “On Some Discoveries of the Methods of Design Employed in Mohammedan Art”, Journal of the Society of Arts, March 17, 1905, pp. 461-465; accessed at http://patterninislamicart.com/drawings-diagrams-analyses/3/methods-design. 13. Aya Johanna Dani Ëlle Durst Britt, “Optical Illusion as a Bridge to Infinity: Escher Meets Islamic Art”, Al.Arte, July 25, 2013, p. 4; http://www.alartemag.be/en/en-art/optical-illusion-as-a-bridge-to-infinity-escher-meets-islamic-art/. 14. Aya Johanna Dani Ëlle Durst Britt, p. 5.15. E. H. Hankin, M.A., “On Some Discoveries of the Methods of Design Employed in Mohammedan Art”, Journal of the Society of Arts, March 17, 1905, pp. 461-465; accessed at http://patterninislamicart.com/drawings-diagrams-analyses/3/ methods-design.16. Faith Gelgi, "The Influence of Islamic Art on M.C. Escher", Fountain Magazine, Issue 76 / July - August 2010; http://www.fountainmagazine.com/Issue/detail/The-Influence-of-Islamic-Art-on-MC-Escher.17. Hankin, Op. Cit.18. Part of an article defining and illustrating hyperbolic tessellations from David E. Joyce, “Hyperbolic Tessellations: Introduction”; http://aleph0.clarku.edu/~djoyce/poincare/ poincare.html:“A regular tessellation, or tiling, is a covering of the plane by regular polygons so that the same number of polygons meet at each vertex. No doubt, the tessellations of the Euclidean plane are well-known to you. They are: {3,6} in which equilateral triangles meet six at each vertex; {4,4} in which squares meet four at each vertex; and {6,3} in which hexagons meet three at each vertex. A notation like {3,6} is called a Schläfli symbol. There are infinitely many regular tessellations of the hyperbolic plane. You can determine whether {n,k} will be a tessellation of the Euclidean plane, the hyperbolic plane, or the elliptic plane by looking at the sum 1/n + 1/k. If the sum equals 1/2, as it does for the three tessellations mentioned above, then {n,k} is a Euclidean tessellation. If the sum is less than 1/2, then the tessellation is hyperbolic; but if greater than 1/2, then elliptic....The hyperbolic plane can not be metrically represented in the Euclidean plane, but Poincaré described ways that it can be conformally represented in the Euclidean plane. One of those is to represent the hyperbolic plane as the points inside a disk. For this representation, a straight line in the hyperbolic plane is represented as the part (in the disk) of a circle that meets the boundary of the disk at right angles....For instance, here is a representation of the tessellation of the hyperbolic plane by pentagons where four pentagons meet at each vertex, that is, the {5,4}-tessellation.

It may look like the sides of the pentagons are curved, but that's just because of the representation we're using. In the actual hyperbolic plane they would be straight. Also, the pentagon in the middle looks larger, but, again, that's due to the representation. You just can't put an infinite plane in a finite region without a lot of distortion.”19. Escher’s hyperbolic tessellation, Circle Limit I, below, with two drawings: one showing the graphing of the triangular hyperbolic plane pattern, and the other showing the central “supermotif” for Circle Limit I. (Douglas Dunham, “Creating Repeating Hyperbolic Patterns—Old and New”, Notices of the AMS, Volume 50, Number 4, April 2003, pp. 453, 454; http://www.ams.org/notices/200304/ fea-escher.pdf--this article contains a complete mathematical explanation for the creation of this pattern.)

Circle Limit I, the Cayley graph of the group [6,4], and the central “supermotif ” for Circle Limit I.

I'd like to thank my friend, Robert Ellis, a mathematics professor at New York University and a union representative for the Adjunct Professors in the UAW, for reading and commenting on this article, and photographer Chris Boundy for the use of his photo.

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